Physics Letters A 372 (2008) 6019–6021

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Physics Letters A

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Orthofermions versus Gutzwiller projection operator approach for the infinite UHubbard model

A.K. Mishra a,b, R. Kishore a,∗a Instituto Nacional de Pesquisas Espaciais, CP 515, S.J. Campos, SP 12245-970, Brazilb Institute of Mathematical Sciences, Chennai 600113, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 9 June 2008Received in revised form 21 July 2008Accepted 28 July 2008Available online 3 August 2008Communicated by A.R. Bishop

A comparison of two well-known approaches for strongly correlated electron systems, namely, nestedBethe ansatz implemented through orthofermion algebra and Gutzwiller projection operator formalism, ismade by calculating the energy spectrum of 1D infinite U Hubbard model for a finite system consisting ofthree particles on a four site anisotropic closed chain. It is shown that orthofermion algebra always leadsto at least an eight hold degeneracy in the energy spectrum corresponding to all 23 spin configurations,consistent with the nested Bethe ansatz solution leading to a 2N -fold degeneracy of energy levels ofan N electron system. Such a degeneracy is absent in the Gutzwiller projection operator approach. Thisfinding shows the limitations of the Gutzwiller projection method and at the same time the relevance oforthofermion approach for the infinite U Hubbard model.

© 2008 Elsevier B.V. All rights reserved.

Gutzwiller Projection Operator (GPO) and Nested Bethe Ansatz(NBA) solutions have been extensively employed to study U → ∞limit of the Hubbard Hamiltonian. In this limit an orbital state isnot occupied by more than one particle. This contrast with the per-missible double occupancy of an orbital state for fermions, whereina spin up and a spin down particle can be in the same orbital state.

As the name suggests, the GPO employs a projection operator toremove a double occupancy at any site. Contrary to this, no suchprojection operator is employed in the NBA approach. Here onesolves exactly 1D Hubbard Hamiltonian employing periodic bound-ary condition for an arbitrary magnitude of the intrasite Coulombrepulsion U . By taking the limit U → ∞ in the resulting wavefunc-tions, the state vectors corresponding to U infinity are obtainedanalytically [1,2].

It has been shown that the NBA wavefunction for the infiniteU is not consistent with the antisymmetry associated with thefermions. In fact these wavefunctions cannot be created throughthe action of a string of Gutzwiller projected fermion creation op-erators on the vacuum state [3,4]. But these wavefunctions can begenerated by the action of a string of the orthofermi creation oper-ators on the vacuum state [3–5]. For orthofermions, a state vectoris antisymmetric only when spatial indices are exchanged. On theother hand, states having different permutations of distinct spinindices σ and σ̄ are independent orthonormal states. This prop-erty is obviously satisfied by any spin system. Any two spin state

* Corresponding author.E-mail address: kishore@las.inpe.br (R. Kishore).

0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2008.07.062

vectors, in which the spins with different orientations appear indifferent order, are independent states. It has been shown that theexact NBA wavefunctions in the U → ∞ limit are consistent withthe orthofermions [3,6]. Thus a comparison between results ob-tained from NBA and GPO approach is same as the comparisonbetween the results obtained for orthofermions and for Gutzwillerprojected fermions.

Here we show that a Hamiltonian expressed in terms ofGutzwiller projected fermion operators and another Hamiltonianbased on orthofermion algebra lead to different results. For thispurpose, we consider the U = ∞ case of the Hubbard Hamiltonian,wherein kinetic energy term alone, subject to ‘no double occupan-cy’ constraint, is present. It is also pertinent to point out that whenU = ∞, all the 2N spin configurations corresponding to an N par-ticle system are degenerate in the NBA approach [1,7].

In what follows, we first provide the set of commutation re-lations for orthofermions and Gutzwiller projected fermions andwrite the U infinity kinetic energy Hamiltonian for the two cases.To compare the consequences of NBA and GPO approaches, asystem consisting of four sites with three particles on a closedanisotropic chain has been studied both within GPO and or-thofermion approach. The energy spectrum and the associated de-generacies have been determined through the exact diagonaliza-tion of the respective Hamiltonian. The consequences of the energydegeneracies in the context of the known exact thermodynamicalresults [4,8,9] for U infinity case are analyzed, and finally summaryand conclusions are presented.

The commutation relations for orthofermions are [5]

ciαc jβ + c jαciβ = 0, (1)

6020 A.K. Mishra, R. Kishore / Physics Letters A 372 (2008) 6019–6021

ciαc†jβ = δαβ

(δi j −

∑γ

c†jγ ciγ

). (2)

Here Latin and Greek indices, respectively, specify the spatial andspin coordinates. In terms of fermion annihilation (creation) oper-ator f ( f †), the Gutzwiller projected annihilation (creation) opera-tor g (g†) are defined as

giσ = f iσ(1 − f †

iσ̄ f iσ̄), g†

iσ = (1 − f †

iσ̄ f iσ̄)

f †iσ , (3)

and these satisfy the following set of commutation relations

giσ g jσ + g jσ giσ = 0, (4)

giσ g jσ̄ + (1 − δi j)g jσ̄ giσ = 0, (5)

giσ g†jσ̄ + (1 − δi j)g†

jσ̄ giσ = 0, (6)

giσ g†jσ + g jσ g†

iσ = δi j(1 − f †

iσ̄ f iσ̄). (7)

The above relations show that g and g† do not form a close alge-bra. However, employing the concept of Generalized Fock Spaces, aclosed algebra for g and g† can be obtained [10].

When U equals infinity, a generic Hamiltonian for three parti-cles distributed on the four sites of a closed anisotropic chain iswritten as

H =∑σ

[t′(a†

1σ a2σ + a†2σ a1σ

)

+ t(a†

2σ a3σ + a†3σ a4σ + a†

4σ a1σ + a†3σ a2σ

+ a†4σ a3σ + a†

1σ a4σ

)]. (8)

The creation and annihilation operators a and a† equal c and c†

respectively in the NBA approach and satisfy the commutation re-lations (1) and (2). In the GPO approach a (a†) takes the value g(g†), which satisfy the algebra given by Eqs. (4)–(7).

In the present problem, the basis set consists of 4 × 23 statevectors because of four configurations from the different locationof a hole on the chain and 23 spin configurations for 3 particles.The Hamiltonian in Eq. (8) does not contain a term leading to aspin flip process. Therefore the problem of finding the spectrumof H gets reduced to the matrix diagonalization in four sectorscomprising of: (i) a 4 × 4 matrix corresponding to Sz = 3/2, (ii) a12 × 12 matrix for the Sz = 1/2, (iii) a 12 × 12 matrix for Sz =−1/2, and finally, (iv) a 4 × 4 matrix for Sz = −3/2. We also notethat since H , S and Sz mutually commute, all the three have com-mon eigenfunctions. Here we concentrate on the energy spectrum.Hence in what follows, we provide only the energy eigenvaluesand not the associated set of the wavefunctions. We sequentiallyobtain the eigenvalues using the orthofermions and GPO approachfor two specific cases corresponding to t = t′ and t = −t′ with ttaken to be positive.

Symmetry of the problem with respect to spin configurationsensures that the matrices in sectors (i) and (iv) and in sectors(ii) and (iii) have common eigenvalues. In addition, if the systemhas a 23-fold degeneracy, matrices in all the sectors have commoneigenvalues. In this case, the problem of generating the eigenval-ues of 32 × 32 matrix gets reduced to the problem of obtainingthe eigenvalues of a 4 × 4 matrix in sector (i). As shown below,this simplification results for the orthofermions approach and notin the GPO formalism.

Using the orthofermion algebra (cf. Eqs. (1), (2)), one gets ineach of the four sectors only three distinct eigenvalues, namely,Eo = −2t , E1 = 0 and E2 = 2t , when t = t′ . The ground energyEo is eight-fold degenerate. The associated wavefunctions are alsoeigenfunctions of the total spin operators S and Sz . To be more spe-cific, the four spin states having eigenvalue 3

2 for S and Sz takingthe values ± 3

2 , ± 12 have no degeneracy in the ground state. On the

other hand, both the spin states with S = 12 and Sz = ± 1

2 are two-fold degenerate in the ground state. This result for orthofermionsis in conformity with the N particle ground state NBA wave func-tions which are 2N -fold degenerate even in the presence of holes[1,7].

When the Gutzwiller projection operator algebra (cf. Eqs. (4)–(7)) is employed for the t = t′ case, energy eigenvalues in sec-tor (i) are −2t , 0 and 2t . These eigenvalues are also obtained insector (ii) along with four additional eigenvalues [11]. Thereforein GPO approach, one gets seven distinct energy eigenvalues atE ′

o = −2t , E ′1 = −√

3t , E ′2 = −t , E ′

3 = 0, E ′4 = t , E ′

5 = √3t and

E ′6 = 2t . The ground state energy E ′

o , unlike 23-fold degeneracyin orthofermions or NBA, now admits only four-fold degeneracywhich corresponds to high spin state S = 3

2 , with Sz taking all thefour values 3

2 , 12 , − 1

2 , − 32 . The low spin states having S = 1

2 arenow excited states.

For the orthofermions, the first excited state having energy zerohas a degeneracy of sixteen, and the degeneracy of the second ex-cited state at energy 2t is eight. In the Gutzwiller approach, thedegeneracies associated with the energies −√

3t , −t ,√

3t , t and2t are four fold each and the zero energy state is eight-fold degen-erate. Both for the orthofermion and Gutzwiller approach, the totalnumber of eigenstates are 32, as expected.

When t = −t′ , the energy eigenvalues in orthofermion case are±√

2t , and each of them has a degeneracy of 16.Whereas the ground state energy −2t is the same for the or-

thofermions and in GPA when t = t′ , Gutzwiller approach gives

a ground state E ′o = −

√2 + √

3t when t = −t′ . Again all the 23

spin configurations are not represented in the ground state. Theground state degeneracy is four and are associated with the spinstate S = 1

2 with Sz taking values ± 12 . The spin states with S = 3

2and Sz = ± 1

2 are excluded form the ground state configuration.

The excited states in GPA are now at E ′1 = −√

2t , E ′2 = −

√2 − √

3t ,

E ′3 =

√2 − √

3t , E ′4 = √

2t , and E ′5 =

√2 + √

3t . The energy levelsE ′

1 and E ′4 each have a degeneracy of 8, whereas remaining four

energy levels are four-fold degenerate.It has been shown that in the absence of magnetic field, the

grand canonical partition function corresponding to the infinite UHubbard Hamiltonian is equal to that of spin-less fermions, butwith the chemical potential μ being replaced by μ̄ [4,8,9], that is,

ZG = Z SFG (μ → μ̄) (9)

with

μ̄ = μ + kT ln 2. (10)

This modification in the chemical potential arises because in con-trast to the energy levels of spin-less fermions, now each singleparticle energy level has a degeneracy of 2N . Obviously, only or-thofermions are consistent with this requirement. In the HubbardHamiltonian, t = t′ . The orthofermion energy eigenvalues for thiscase are −2t , 0 and 2t . As shown earlier, the first and last eigenval-ues are eight-fold degenerate and the middle one has a degeneracyfactor 16. These eigenvalues correspond to the Bloch states for afour site problem with k = −π , ±π/2, 0, respectively. Time re-versal invariance implies that E(k) = E(−k) [12]. This explains the16-fold degeneracy of the zero energy eigenvalue, and makes en-ergy levels corresponding to each k eight-fold degenerate.

To summarize, a study of three particles on a four siteanisotropic closed chain have shown that the orthofermions andGutzwiller projection operator formalism lead to different energyspectrum. Whereas the former approach is consistent with thereported 2N -fold degeneracy associated with an isotropic closedchain, shown by the exact NBA solutions of the infinite U Hub-bard model, Gutzwiller projection operator does not reproduce

A.K. Mishra, R. Kishore / Physics Letters A 372 (2008) 6019–6021 6021

this exact result. This shows the limitation of the Gutzwiller pro-jection operator and simultaneously highlights the equivalencebetween orthofermion and the nested Bethe ansatz formalism forthe U → ∞ limit of the Hubbard Hamiltonian.

Acknowledgement

A.K. Mishra acknowledges the financial support from CAPES,Brazil.

References

[1] M. Ogata, H. Shiba, Phys. Rev. B 41 (1990) 2326.

[2] H. Shiba, M. Ogata, in: G. Baskaran, A.E. Ruckenstein, E. Tosatti, Yu. Lu (Eds.),Strongly Correlated Electron Systems II, in: Progress in High Temperature Su-perconductivity, vol. 29, World Scientific, Singapore, 1991, p. 31.

[3] A.K. Mishra, Phys. Rev. B 63 (2001) 132405.[4] R. Kishore, A.K. Mishra, Phys. Lett. A 369 (2007) 226.[5] A.K. Mishra, G. Rajasekaran, Pramana J. Phys. 36 (1991) 537.[6] A.A. Zvyagin, Low Temp. Phys. 33 (2007) 448.[7] P. Fulde, Electron Correlations in Molecules and Solids, Solid State Sciences,

vol. 100, Springer, Berlin, 1995.[8] D.J. Klein, Phys. Rev. B 8 (1973) 3452.[9] G. Beni, T. Holstein, P. Pincus, Phys. Rev. B 8 (1973) 312.

[10] A.K. Mishra, G. Rajasekaran, Pramana J. Phys. 45 (1995) 91.[11] P. Fazekas, Electron Correlation and Magnetism, World Scientific, Singapore,

1999.[12] J. Callaway, Energy Band Theory, Academic Press, New York, 1964.